


Manifold

by pendrecarc



Category: The Magnus Archives (Podcast)
Genre: Epistolary, Existential Horror, Gen, Geometry, Mathematics RPF, Statement Fic
Language: English
Status: Completed
Published: 2020-03-27
Updated: 2020-03-27
Packaged: 2021-03-01 03:40:28
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 2,328
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/23278738
Author URL: https://archiveofourown.org/users/pendrecarc/pseuds/pendrecarc
Summary: "I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe."János Bolyai, in a letter to his father
Comments: 11
Kudos: 19
Collections: Worldbuilding Exchange 2020





	Manifold

**Author's Note:**

  * For [kimaracretak](https://archiveofourown.org/users/kimaracretak/gifts).



Archivist’s Note:

János Bolyai was a Hungarian mathematician of the 19th Century. He is principally known for his work on non-Euclidean geometry; that is, geometry that doesn’t follow logically from all five of Euclid’s axioms, or fundamental rules. When Bolyai was a young man, he began to explore geometries that might be possible in a world where Euclid’s fifth axiom (also known as the _parallel postulate_ ) does not hold true.

The parallel postulate may be explained as follows:

It is possible to create two infinite lines at a constant distance from one another.

Or, in layman’s terms, if a railroad ran on forever, the two tracks might never grow closer together or farther apart.

This statement seems like an obvious fact, but mathematicians had disputed it for centuries before Bolyai chose to reject it. As a young man, he began to construct systems where two infinite lines must eventually join or diverge. Writing ecstatically to his father in 1823, he said, “out of nothing I have created a strange new universe."

The attached document is a translation of a letter discovered among his papers. It is also addressed to his father, another prominent mathematician of the day. While undated, it appears to have been written between Folkas Bolyai’s death in 1856 and János’s own in 1860.

********

Dear Father,

I write to you out of habit, you who are several years dead. There is no-one else to speak to of what I’ve discovered. Gauss is lost to me, too; but I’ve heard of a young man who studied with him, a man named Riemann, who I think might be brought to understand. I am told he has proposed a complex system of infinite geometries—but he has not published his findings. I can’t tell if his failure comes of cowardice or circumspection. I hardly know which of these I find within myself.

Well do I remember your reaction when I first set down this path. "You must not attempt this approach,” you wrote so long ago. “I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life.” As though the fifth of Euclid’s postulates was necessary not only for an understanding of the physical world, but for the preservation of your immortal soul! I laughed at the time, and won you over, and made my proofs public; and neither that physical world nor my mind began to unravel as a result. Now I begin to see new value in my father’s wisdom and to regret the impulses of my youth, as I suppose all men do when they reach a certain age and accept the reality of their deaths. Yet it is not the prospect of death that appalls me. I hesitate even to use the word “reality”.

I was working late three nights ago, when I closed my eyes against a headache and saw a vast great plane stretched out before me like the page on which I had been writing. I began to walk across it, thinking I had only fallen asleep. I amused myself by wondering if it was a flat Euclidean plane, or if, when I continued long enough, it would begin to curve under me.

After some time I realized I was not alone, and that another figure paced along beside me.

It looked, so far as I could tell, like a man dressed in strange clothing in a fashion I did not recognize. He walked at some distance so that I could not see his face, moving neither faster than I nor very much slower. I tried after some long minutes of this to moderate my pace, as an experiment, and found he adjusted his own speed to match it. Together we traced out two parallel lines upon that plane.

"Do you suppose we will ever meet?" I ventured with a laugh; this is not the sort of wit that serves me at dinner parties, though it might be indulged in the privacy of my own mind. But the laugh died in my throat. The sound fell odd and hollow in that space, and it made me uneasy to hear.

He turned his head toward me, still walking, and asked, "Should you like to?"

His voice, too, wrenched oddly at the ear, queer and distorted. "If you would, sir," I said with a bow. A figment in a dream may not warrant such courtesy, but you see, Father, that I have grown out of the choler you once deplored, and have less need now to give or take offense.

He bowed his head so deeply that I suspected him of irony, and he continued to walk with me.

I could not tell you how long we kept pace with one another, never deviating from our original course or turning deliberately in either direction. At length he began to appear more clearly in the corner of my eye. At first he was in profile, and I thought what a strange profile it was, all unlikely angles and odd, jerking movements of the limbs. And then his face was inclined toward me, and I knew we must be growing closer to one another, for I could see the great height of his nose, the jutting promontory of his chin, and the hollows like ocean depths between them.

"Are you unwell, sir?" I asked, in no little alarm. It looked to me like a face that had suffered under extreme privation, and I could not imagine how a man could wear it and live.

"You are kind to ask," he said. By now I could see the curve of his mouth and began to wish I could not. "I am very well indeed. Could there be anything better than to walk together in this way?"

I knew, from the fact that we had drawn closer, that the plane on which we walked was not flat like a tabletop. Now I saw and felt other evidence of this, as the ground beneath my feet began to arch away. With every step I leaned forward to cover more of that ground, and it seemed to recede as I reached for it. I reeled on, falling more than walking, as even the air around me began to stretch and thin and pull me with it. The skin of my face felt as though some force beyond the horizon was tugging at it. My eyes began to bulge in their sockets, and as my chest expanded with every desperate breath I thought I would not be able to expel it again.

Yet still I walked until that man was not beside me but before me. Then he looked me full in the face and smiled.

I woke to a guttering lamp with my cheek pressed against an ink-spattered page. _That_ page, as far as I could tell, was perfectly flat.

It seemed a whimsical sort of dream for a geometer to have, and at first I was inclined to laugh it off. Then as I breakfasted the next morning I looked at the convex back of my polished spoon, and my eye began to slide nauseatingly down its side until the thought of food became distasteful. All that day there was a queer irregularity in the walls of my home, the angles of the furniture on which I sat, the shape of my hand as I held it before my face. I went out into the sunlight, but even the birdsong fell warped upon my ear.

I did not stay late at my desk that evening, but went early to bed. Yet I dreamed of that place again.

This time when I met him I said, “I thank you, sir, for showing me the joys of one sort of geometry; but tonight I would prefer it if we were not to meet.”

I had not turned my face, and I could not see him as more than a faint, unsettling shape at the edge of my vision. Yet I thought he smiled. “Never at all?”

“No,” I said firmly. “I should prefer it if we walked away from one another.”

“That can be arranged.”

That night, about the time our perfectly straight paths began to diverge, he began to speak to me. He asked me to consider the consequences of this sort of geometry on our physical world. I imagined a great ship of the previous age carrying a brave press of sail, pointed unerringly toward the horizon. Her captain walked the deck every day at noon, sextant and compass in hand, and plotted out his course with great skill and accuracy. At length land was sighted in the distance. I saw, though he did not, how the ship began to heel, her prow tilting up under the rising water. He took reading after reading as days and months passed, as all the hundreds of souls aboard her grew gasping thin and thirsty. Reading after accurate reading, so that his course was true, his location certain, but the destination never grew closer. The ship ran nearly vertical, the sea streaming down her sides as the crew dropped dead one by one. Still the captain stood at the rail, eyes fixed forward as his face became dark mummified leather and his body withered away, looking out toward the ever-receding horizon.

As I thought on this my companion continued to speak, though by now we had been walking away from one another for what seemed hours or days. His voice was distant, echoing oddly, but quite clear. He bid me think now on the course of a railroad-track or to imagine the flight of a hot air balloon. I became aware of an odd shuddering through my wrists, turned one to look at the underside, and saw there the pale blue channels of my veins, the outline of the walls holding in my blood under the skin. And I saw that within the confines of my wrist these walls had begun to diverge, to bow outward—each beat of my pulse forcing blood out into a vessel that did not narrow at its extremity, but expanded. I did not see how my flesh could hold it or my heart bear to carry it.

I turned to look over my shoulder, about to call out in a panic, and saw at a great distance the retreating back of my companion—and between us not the flat surface on which I thought I had been walking but a great, deep round cavern, and I walking up its side. I gasped, and he laughed at me. The sound did not come to me from behind, but from before and above. I felt the grip of my feet on that surface begin to give way. Closing my eyes, I turned my face forward and continued walking, ever up and away, the echoes of his laugh growing nearer and nearer.

Again, I woke to a world I thought I knew. I spent most of that day in the garden with sunlight pressing on my closed eyelids. I hated to open them, for when I did the path before me would warp and stretch, and I could not keep my footing. Once a bee buzzed in my ear, an odd, strobing sound, before it came to rest on my knee. I dared to look at it and saw its glassy round eyes begin to curve around in the wrong direction, like great black bowls into which I might easily fall. Each faint twitch of its long, translucent wings threatened to split the sky open above me.

Last night I sat at my desk before a blank page and waited for him to come.

“How shall we walk tonight?” he asked me, all pleasant courtesy.

I had prepared my answer. “On a flat and ever-fixed Euclidean plane, if you please.”

He began to laugh again. “Oh, but I do not please! Where is your sense of discovery, geometer? How dull, to walk along together, never parting, never meeting. How _predictable_.”

“Then which of your alternatives must it be?”

“Both,” he said. “Neither. We have discarded the fifth of those axioms your profession values so highly, but that is no great sacrifice; you chose that yourself, years ago, without regret. Let us see what can be done without the others. Tell me, geometer, what is the first of these?”

In dread, I told him, “Euclid’s first axiom states that between any two points, there may be drawn a straight line.”

“That,” he said, “is a lie.”

And the world came apart.

I hardly know how to put it to paper. There is no language for the total dissolution of space, no notation for the impossibility of a circle or the inconstancy of a right angle. I have been trying and failing these many hours to create a diagram that can capture it. Each time I think I have achieved this, I see that each clear line of ink on the page refused to remain where I put it. My hand releases the pen and finds it held even tighter in my grasp. The bones of each finger turn in upon themselves. My eyes will not focus on what is before them, but on what is behind and above and below and _inside_. Even this letter has been a monumental effort. I hardly dare hope the words have any form to them, much less any sense. You see I have tried to scrawl a proof of my derangement in the margins, but the world is too small to contain it.

The hour grows late; my hand is tired. I will not take to my bed. One place is as good as another, when the space between them is both infinite and infinitesimal.

I do not know what he will show me tonight, but it is too much to hope my mind can withstand it.

In affection and terror,

János


End file.
